Calculating the relative likelihood with AIC values I'm using AIC for model selection, and would like to use a relative likelihood measure to quantify how many times a model with minimum AIC (AICmin) fits better than the alternative (with AICi).
For that, I'm using Burnham et al. (2011) formula, which is:
RL = exp ( 0.5 * ( AICmin - AICi ))

The expression is quite easy. However, I'm worried to miss something. In mi case, AICmin is negative (AICmin = -239.10, AICi = 210.43), which makes the difference term (AICmin - AICi) also negative, and thus a relative likelihood on the order of zero (RL = 2.43e-98) and does not make sense.
In the original article I don't find any reference saying that the difference should be absolute, but if so, the ratio becomes too high (RL = 4.11+97) to me to feel sure. Am I missing something? Thank you!
 A: The issue is that this term "relative likelihood" isn't a likelihood at all. The AIC is both a negative log-transformed likelihood (so that lower is better) as well as scaled by a constant related to the number of parameters.
The formula for the AIC is:
$$AIC = 2K - 2 \log(L)$$
It's not surprising to see a log likelihood that is negatively valued, or an AIC that is negatively valued.
In spite of all of that, it's still a separate issue that the relative "likelihood" is close to 0. All the RL is doing is transforming an AIC back to a quasi likelihood ratio. A ratio that's close to 0 means the denominator AICi is very, very big compared to the numerator AICmin. That's not surprising. All it says is AICi really sucks compared to AICmin.
A: Have you considered AIC weights? They serve to normalize the relative likelihoods from a set of candidate models. Also, AIC weights allow quantification of the relative probability that a model is correct (relative to the other models considered) for the given data. See Burnham and Anderson (2002) for more info.
Burnham, K.P., and D.R. Anderson. 2002. Model selection and multimodel inference: a practical information-theoretic approach, 2nd edition. Springer-Verlag, New York, NY. 488 p.
